Let $G$ be a $p$-group contained in $S_{p^m}$ for some $m\in \mathbb{N}$, this is, the symmetric group of degree $p^m$. Assume that the lower central series, LCS for short, of $G$ is well understood.

Is there anything known about the LCS of $W(G):=G\wr C_p$, where $C_p$ is a cyclic group of order $p$, and $\wr$ is the (right regular) permutation action of $C_p$ on $G\times \dotsb\times G$. That is, if $(g_1,g_2,\dotsc, g_p)\in G\times \dotsb\times G$, then the action of $C_p=\langle x\rangle$ on the base group $G\times \dots\times G$ is defined by $(g_1,g_2,\dotsc, g_p)^x=(g_p,g_1,\dotsc,g_{p-1})$.

I am particularly interested in the indices $|\gamma_i(W(G)):\gamma_{i+1}(W(G))|$ for $i \in \mathbb{N}$.

I have been searching for some bibliography or papers on the topic but I have not been able to find anything.